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A version of the binary Darboux transformation is constructed fornon-stationary Schroedinger equation in dimension $k+1$, where $k$ is thenumber of space variables, $k \geq 1$. This is an iterated GBDT version. Newfamilies of non-singular and rational potentials and solutions are obtained.Some results are new for the case that $k=1$ too. A certain generalization of acolligation introduced by M.S. Livsic and of the $S$-node introduced by L.A.Sakhnovich is successfully used in our construction.

Time-dependent Schroedinger equation in dimension $k+1$: explicit and rational solutions via GBDT and multinodes